Interconnection and damping assignment passivity-based control is a technique that regulates the behavior of nonlinear systems assigning a desired (port-controlled Hamiltonian) structure to the closed-loop. Since the introduction of this controller design methodology five years ago (1999), many theoretical extensions and practical applications have been reported in the literature. The theoretical developments include some variations and shortcuts that are useful when dealing with particular classes of systems, and the incorporation of additional features to handle control scenarios other than just stabilization. On the application side the method has provided solutions to a wide variety of physical problems. The purpose of this paper is to review the fundamental theory of this control system design approach. Main new results and practical applications as well as current open problems and future directions are discussed in a companion paper.